Consider the functions and .

MATHEMATICS · Multi-part · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Consider the functions

y=\tan x

and

y=\sec x

. (i) Explain why every real number

x

satisfying

\cos x = 0

corresponds to a vertical asymptote of each of these functions. (ii) Using the periodicity of

\cos x

, describe the complete set of vertical asymptotes of

y=\sec x

in words. (iii) Show that between any two consecutive values of

x

for which

\cos x = 0

, there is exactly one value where

\sin x = 0

. How does this fact determine the positions of the vertical asymptotes of

y=\csc x